Subcovers and Codes on a Class of Trace-Defining Curves

In this paper, we construct some class of explicit subcovers of the curve <inline-formula> <tex-math notation="LaTeX">$\mathcal {X}_{n,r}$ </tex-math></inline-formula> defined over <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{n}}$ </tex-math></inline-formula> by affine equation <inline-formula> <tex-math notation="LaTeX">$y^{q^{n-1}}+\cdots +y^{q}+y=x^{q^{n-r}+1}-x^{q^{n}+q^{n-r}}$ </tex-math></inline-formula>. These subcovers are defined over <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{n}}$ </tex-math></inline-formula> by affine equation <inline-formula> <tex-math notation="LaTeX">$g_{s}(y)=x^{q^{n}+q^{n-r}}-x^{q^{n-r}+1}$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$g_{s}(y)$ </tex-math></inline-formula> is a <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-polynomial of degree <inline-formula> <tex-math notation="LaTeX">$q^{s}$ </tex-math></inline-formula>. The Weierstrass semigroup <inline-formula> <tex-math notation="LaTeX">$H(P_\infty)$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$P_\infty $ </tex-math></inline-formula> is the only point at infinity on such subcovers, is determined for <inline-formula> <tex-math notation="LaTeX">$1 \leq s \leq 2r-n+1$ </tex-math></inline-formula>, and the corresponding one-point AG codes are investigated. Codes establishing new records on the parameters with respect to the previously known ones are discovered, and 108 improvements on MinT tables are obtained.

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